Resolution of holomorphic functions by distributions.

Resolution of holomorphic functions by distributions,

Serre [5] proves the following duaU.ty theorem:

Let X ·be a compact complex manifold, dim X "' n, and let \'/ be a holomorphic vector 'btmdle on X, and

vl*

the dual bundle of W.

Then the vector spaces Hp'q(X,i'l) = Hq(X,(?P(w)) and

w-p,n-q(X,.W*) =

W

1-q(X,(?n-p(vl*)) are (canonically) dual to each other, in particular, they have the same (finite) dimension,

To prove this, he resolves the sheaf(? P (i'l) of germs of holomorphic p-forms with coefficients in a holomorphic vector bundle W in two (fine) ways:

•••

^{... 0}

o

..,cl<w)

..,~,(p,o)(w) ~D'(p,1)(lv)li" -

^...

o'

where,A(p,q) (vl) is the sheaf of germs of

<f'-

forms of type (p,q) with coefficients in W, and

J)

¹ (p, q) (vi) is the same kind of dis-

tributional forms,

~1us one can calculate Hq(x,8P(w)) from either sequence, Since

!i)'

is dual to~ "'

<f'•

this is a natural procedure. The above result is a consequence of the well-lmown Grothendieck lemma, and of the faot that i f TE~)' (u), U open in vn, satisfies

oT/ozj

=

^{0 for j}

=

1, ••• ,n , then T is a holomorphio function, Concerning the Grothendieok lemma for distributions, Serre refers to a paper of Dolbeault [

1],

in which this distributional

Grothend.ieok lemma is stated, However in this paper Dolbeaul t gives no proof, but says that this is an unpublished result of Grothendieck.

We propose to give a proof of the distributional Grothendieok lemma below, The proof is modelled on the method in Narasimhan [4], According to Narasirahm1, this is the method of Grothendieck, as exposed by Serre. 'ITo also prove the statement about tho distri- bution T above, and this proof is a generalization of the 1 - dimensional proof in Gunning [ 2],

We first need the following lemma,

Let L^{1 ,} K, L be compact su·bsets of :m.s, V, QJn respectively, Denote by (t, z, w) points in IRs ^>< 0 X

qp,

_{Let gE~'(U),}

where

u

is an open subset of ^{!IRS X (l X}

<P

containing

L'

^{X K X}

L,

and suppose

og/b ~c

"'

o

^for 1 <k<n

'

^where

- --

w "' ( w' , , , , , w^{11) ,} Then there is a distribution fE ¹ (U ^{1 ) ,} where U' is some open set contained in

u

and containing L' _" ^v K >< L such that of/o ;)c "'

o

for 1 <k<n and

' - -

of/o z

""

^{g in U I t}

Proof: For simplicity, we assume that s ""0 , as this does not affect the proof, We may also suppose that gE ^{€ 1}(QJ X ~n), since

we can mul "biply g by a C⁰⁰

0 function having support in

u

and being equal to 1 in a nbh, of K >< L , and then oonsidel' g in this last nbh,, Thus we assume g is a compactly supported distribution in C X

4fl •

The proof is somewhat technical, and we divide it into three parts:

I) A statement needed fo define a distribution f in II), II) Define f and prove that f is a distribution,

III)Show that f is tho di.3tri"bution we seelc,

I) For cp E cf"' (<ll x <lln), let 'fo.cpEcf"' (m X

~)

be the translation by oEQJ x I!Jn thus ( 'fcr,~o) (z, w) "'cp((z,w)-x).

ro oh ~

hEC₀ (V) ¹ and we have ~(s) "'-(g , 'f(s,o)

oz)•

We checlc this: I f 0

+

^{t ER,} then h(s+t)-h(S) "' (g , t

( 1 )

Now lim t-70

1-(!p+t,o) cp)(z,w) -(r(s,o)cp)(z¹w)"' t

lim ce(z-ip-t,w)

t cp(z-)~·ll"'- ~XW(z-s,w)

^{= -}

C (~p,o) ~X)(z,w),

t-70

where Z = x+iy , with

x,

^{y E R •} Similarly , i f 0

+

^{t E JR,}

(r( ) cp)(Z,w)- ('f(t;,,o)'PJ.(z,\'1)

then lim g+H,o .

_n ___ _

"' - (r

('R,o) ~)(z,w)

^•

We now prove that lim t-7o

t

and

r( . ) c,o-or )cp · (2)

~?Ht,zO (~,o

^{"'- 'f (s,o)}

~

^t for t Em ' that is that these limits hold in the Frechet space C ⁰⁰(0 >(Gf).

Consider the first limit. Let m be a non-negative integer and A a compact subset of V X

QP •

Denote for the moment b;y D

a differentiation mononial in the real coordinates of 0 x On , and by

II IIA,m

the semi-norm in 0 =(ox on) given by A and m, using monomials of order :::; m, Then lit (T

(~+t,O)

Cfl-'f (s,O)cp) + 'f(~,o) ocp/oxiiA,m

= sup sup 1

D(~

(T (t;;+t o) cp-T (s o)cp) + T (t! o)ocp/ox)(z,w)

I·

ord

D:::;

m (z,w)EA ' ' "'

Consider for instance the case D

=

o/ouk, where wk "' uk +i

~

with

1 k k 2 k

= <t

(7(s+t,o) ocp/ou - T(;,o) ocp/ou ) + T(;,o) o cp/oxou )(z,w) , Similar expressions hold for o/ovk , o/ox, o/oy , and other mono- miaJ.s ^{D •} All these expressions tend uniformly to zero on the compact set A as t tends to zero, Henc.e the first limit expres- sion in (2) above, and similarly also the second, is true, By the above expressions we therefore get, since g is contiuous on

0 co(Ql X Qln) ^I that oh/O~i 1 and oh/os2 exist, where

s =

^r.;₁ + ig 2 with s₁ ¹ s 2 E IR., and further

This gives oh

i;i

^(s )

=-

<g ' 'f(g,o) ~ ben ) , and also, when applied

several times, The

last shows that h is

<fO•

1'/e must show that h has compact support, Choose R ) 0 and a compact set B in On such that supp g u supp cp c KR >< B , where KR "' {

I

z !:::; R } •

I f !si>2R and (z,w) EKn >; B, then 1~-z1::.: lis!- !zll=

Is 1-

lz!>R, thus (z - ;,w) qKR x B , Hence supp g

n

supp T (s,O)cp = ~

for Is 1 > 2 R , and thus also <g , T (t;;,o)cp> "' O for Is I>2R, which gives supp h c K2R •

II)

wher·c Y11 is Lebesgue mcaam:e on C , Ther, fE

0

^{1 ( (]) X IDn)}

We checlc this:

B I)

.Y (f,~) is well-defined.

(3)

j_s integrable over C),

Clearly, f is linear, To prove that it is continuous, it suffices to prove that it is continuous on ~RXB (ID x~¹) , by properties of LF spaces

[Tr~ves ^[6]],

where KR"' {lzl:::; R},

B

compact in

~

•

^\'le can also talce Kll XB so big that supp gc:KR XB , as in I) above, As in I) supp (g~(g,T(~,O) ~))c K2R for supp

cpcKR X B , Introduoingpolar coordinates, (r,e)

'

^on

c ,

\ve get

0 0

2TT 2R

<.:Lj' J'

- TT . where

0 0

i(g,T (t;,o) >I attains its maximum on { Is I:S2R} "' K2R at soEK2R.

For sEK2R we have supp T (

s,

^{O )~ c K}₃R X B , and the continuity of g gives for some constant C > 0 that l<g,T(s

0

,o) ~)

I

<

c

^{sup sup}

D (z,w)EK 3RxB

where D means

differentiation monomial in real coordinates of ~ X ~n , and the D--sup is taken over monomials of order less then some integer,\Ve have

I(D

cp) (z- ~

0

^,w)

I::;

sup j(np)(z,w) 1₁

(z1w) EKR XB since supp cp::KR X B , Thus

I

^{(D cp)(z,w)}

I•

By the above we then get

=

sup D

sup B

I

^{(np) (z,w)}

I

for cpECK....xB (<!l

x

ofl) , which proves

(z1v!) EKR X K

continuity, since sup supj (D:p) (z,w)lis a semi-norm on D (z ₁ w) EKJi(B

C ~J{<B ( ^(J)

x

~n) ^• (Order of D is 'botmded).

III) Remember nov/ that in the beginning of the proof we multiplied the original g , call it here g

0 , by a ~ ^(C

x

en) - Function with support in the given U and being equal to

1

in a n'bh, U ¹ of K X L , More accurately we do this as follows: Let U, be open in

a

^and

u

2 open in Oln , such that KXL c

u

1

xu

₂ ^{c: U ,}

and let e

1

^E~

^(u

^{1) and}

^o

^{2 E ~}

^(u

^{2) be suoh}

^e

^{1 ~}

¹

in a nbh, of K and e

2 =

1

in a nbh. U¹

2 of L. Then let eE~ (U) be 0 (z,w): =

e

1 ^(z)•e2(w) • We have then

e

= 1 in U1 :

= ^u

₁^{1 X U} _{2 •} ¹

\'le talce our g as g: , ⁸ g

0 , I f cpE ~ ( ^{C X} <fl) has support in C >(

U2 ,

then we get (og/o~k , cp) ^<=

<e1

^{- g oe2}

o;;"f

^{0 , cp)+}

og

og

;o;;l'

⁼

o

<818

2

-k ^,

^cp)

⁼

⁰

_'

^sinoe and since

ow ⁰

= 0 in U¹

2 •

With g constructed in this way, f is the distribution we seek,

Check: Let cpE~ (U^{1) ,} Then for any sEC we have supp

'f(g,o)

^cpc: C

x u2 ,

and the above gives: (of/ovf- , cp)=

- (f, ocp/owk

>

=

1 ^J.

^(g,

'f (g,o) ~ ^>

^dm

^(g)

TI . ~

1 r·

=

Ti J

c c

<

_{og ow}

I

^-k _'

'f(!;,o)

cp) c1m

(g) =-1j'2dm

T1 g (g) = 0 •

c

Thus

=

^{0 in U1 ,} which is pe.rt of what we need,

for cpE~ (U¹⁾ 1 we have (of/oz,cp) =- (f , ocp/oz )

1

j'

= -TT·

c

.2s! £S2

<g, 'f(>o) o'Z > 1 J (g,'f(g,o)

oz

>

S

^c1m (g) = TT { 1

S

^c1m (g)

c- O)

tr (g, 'f(g,o)

cp)dm

(g) ,

^{by I), Since}

we have

(4)

Further,

1

1 .

¹

=- 2ni sd ( g, r (s,O)cp) 1\ ds • 0 -·{0}

Nmv i f a E 0⁰⁰(0), then in 0 ·- {0} 1ve have d(as-1ds)

= d(ar;-1) "ds = r;-¹da Ads-a.!f2dsl\ds = r;-1do:Ads • In the above cal- culation this gives (of/oz,cp)

=- 2 ^~i J d((g,r(r;,o)cp)~s)

0-{0}

=

lim-2~1 f

d((g,r(s,o)cp>flO =lim

2~i J (g,r(s,o)cp)~s

8 ....

o

²ⁿ

I

^r;

l:::.e

^{8 ....}

o I

^r;

I

⁼⁸

1

J.

=lim 2^{ri (g,r iB cp)dB =} (g,r(O,O)cp)=(g,cp),since g .... ~,r(s,O)cp)

8 .... 0

0 (ee

,o)

is continuous. vie see now that of/oz = g in U^{1 •} Since this g equals the original

lemma is proved.

g, called g

0

We no1v need some notation

in part III), in U¹

,

the

For

u

open in on ' let D I (p' q) (U) be the forms of type (p' q) vJith distributional coefficients ("currents .. ). Thus w E D' (p,q)(U) can be written w = L: aiJ dzi 1\ dzJ, with aiJ E D¹ (U) , I and J

I,J I

multi-indices of length p and q respectively, and dz =

i1 ip

dz 1\ •• , 1\ dz The operator

a

i f

acts as usual:

etc.

E

~

dz) "dzi" dzJ • IJj

azJ

We no1v have Grothencieck ¹ s lemma, the proof of which is found in Narasimhan [ L~] ^• (In the

cF-

^case) Since it is short, vTe give it for completeness,

Gro~hendieck's le~na

Let K1, ••• ,Kn be compact sets in 0 and

s =

K1 x ••• xi~co. ⁿ

Let

w

E ;0¹ (p,q)(U), 1·1here q ::: 1 and

u

is a nbh. of

s .

^{I f}

ow

_{= 0 '} then there exists

w'

^{E JjJJ} (p' q-1 ) (U) such that

ow'

= w in a nbh. of S •

Proof: If 1 < v E 2Z, let Ap,q = Ap,q(S) be the space of ele-

v \)

ments w E $J ^{1 (p' q)} (U) defined in a nbh. U = U(w) of S and such that w does not involve -v -n

dz

,o ••

,dz • Thus IIJ

=

I -J

E aiJ dz Adz IJ

where I f v > n then

'

J = ( j 1 , ••• , j q) with 1 ~ j 1 < ••• < j q ~ v-1 • is the space of elements lU E ,GJJ¹ (P,c({u) defined in a nbh. U = U(w) of S • Also if w E A~,q and q ~ 1 then clearly w = 0, and thus the lemma is trivial if w E A~,q • Suppose the lemma is true for all

w

^{E ~,q, and}

E Ap,q-1

\) - \ )

\,re can write w = dz Aw 1 +w

2 , vlith w 1

let w E

Since I f ow =

o ,

then - dzv A ow

1 + ow

2 ⁼

o •

involve dzv, ••• ,dzn, we have ow

1/ozj = 0 and

w1 and w2 ow2/ozj =

Ap,q

\1+1 • E Ap,q

\)

.

do not 0 '

( componentwise differentiation), for j = v + 1, ••• ,n • By our first lemma there exists x' E

~J(p,q-

¹

)(U)

in a nbh. U¹ of S, 1vith ox' /ozj =

o

^{for j} = v + 1, ••• ,n and ox ¹ /ozv = w

1 • !'1ul tip lying x ¹ by a ~(U')- function 1vhich is equal to 1 in a nbh. of S, v1e see that there exists x E

$

t (p' q-1 ) (en) 1-1ith

ox/oz)

^{= 0 for}

j

=

v+1, ••• ,n and ox/ozv

=

^w₁ in a nbh. of

s.

This implies that w- Cix E A~' q • Since

a

^{(w-ox) = Ciw =} 0 , there is, by the induction hypothesis, an element w E

9J

¹ (p,q-1 )(Cn) .vith w-

ox

= Cil)i in a nbh, of S •

We further need the following theorem, a proof of .vhich in the case of a Riemann surface can be found in Gunning [2]. To generalize that proof to the case of arbitrary dimension, vie need a Cauchy formula in several variables. Since v1e .vill use differential forms, Stoke ¹ s theorem etc., i t is convenient to use the Cauchy--!'1artinelli formula. This reads as follo.vs: If f is a holomorphic function in a nbh, of

s

^{+ KR c} en, .vhere KR = (z E en

l

lzi.:::_R}, then

f( ~)

n\n-:-1)

= (-1)

2

.Cn-1)!

J

^f(z) w(z-S), vrhere (2ni )n S+S J z-s 12n

R

and w( z) := ^{n (} _{2: --1} ^{)k -k} z dz A •• ,Adz A dz A •• ,Adz A ••• Adz ^{1 n -1}

---R

^-n (A means k=1

"omission", as usual) There is a simple proof in Lang [3].

The generalization of the theorem in Gunning is:

Let 'I' E ~~ (U) ₁ where U is open in en ₁ and assume that

o'I'/ozj

=

⁰ for j

=

1, ••• ,n. Then T is a holomorphic function in U •

Proof: Rewriting the and

s ,

and putting

eauchy-Martinelli formula, interchanging n(n+1)

A:= (-1) 2 (n--1)! 1ve have f(z) =

(2ni )n '

z

A

J'

f(z+r.;)ill.l for

1~12n ^f holomorphic near z + KR • In particular,

~

for f

=

¹ we get 1 = A

J ~~~~~ .

^{If f is only}

^cF

in a nbh.

SR of z , we get then:

lf(z)

-As·

f(z+s) wCS)I = lA

J

f(z) w(s) -A

J

f(z-:-s) w(s)l

~ ~ ~

~ SR SR

= lA

J (f(z)-f(z+s))~~~~~~.

This quantity can be made arbitra- rily ~11 by trueing R small enough, since f is continuous.

Thus we see

f(z) ⁼ lim A

J ^f(z-:-S)~

R->0 1~1 n

~

for f

cF

near z '

(5)

To prove that T is holomorphic it is sufficient to prove tha·t; it is

cF,

and to show that, we will write any cp E e~(U) in a spe- cial form so that we can use the conditions given on 'I' • Let then, for e > 0, U

8 := (z E en

l

dist(z,en- U) > e} , and let cp E

cFcu )

_{c E:}

c ~(U) • Further let p E ~(en) be such that p(s)

=

1 for

Is[< e/2, and supp p c (isl<e}. By (5) vre get, for z E Ue, cp(z) = lim A

J

^cp(z+S)

^~w(S),

^{since p = 1} on SR for R < e/2.

R->0 lsi n SR

By Stoke we get, since the orientation of SR is ouhmrd, observing that

s ....

f(z+s)p(s) is defined for all s E C since

z E

u :

cp(z) = - lim A

J

d[cp(z+S) p(s) w(s)J (6)

e R->0 isi>R lsi2n

Here the d is 1v,r.t. s • We have d[cp(z+s)

p(~)

^{w(s)J =}

lsl-n

dcp(z+s) A· o(s) w(s) + cp(z+s)d[__E_{s) w(s)J • Since each term of w

~ isi:!:n

contains ds'A,.,Adsn, we see dcp(z+s) A p(S) w(s) = <Jcp(z+s) lsl2_n

A~

^{w(s) =}

~

ocp(zi;S) dsj

A~

^w(S)

lsi j=1 0gJ lsi n

=

~ ~ Js2..cz+s)·~·(-1)kskdsjMs'A-.MsnMs'A

... M?A ... Msn j=1 k=1 ozJ Is[ n

=

~ ~(z+~)

P(s)

uj·(-1)n-

¹

d~'A Ad~nAdP.'A Ad~n

. -J "

I

12n " " .. • " " • • • "

J=1 oz s

n . .

n-1+n(~r1)

= 6

~(z+s)~

^SJ•(-1) ds'Ads'A ••• AdSnAdSn j=1 ozJ lsi n

1 n(n::t1) n

n·- + (s)

= (-1) ·(2i)n. 6

~(z+s)·~~·sjdm(s),

^{where dm is}

j=1 ozJ [si n

_...._,

Ad'.::nAdS'A,,,AdSJA,,,AdSn

n-1+ n(n-:-1) ^~

= (-1) 2 -(2i)nh(s) dm(s),

this together, \ve get d[cp(z+S) p(S) w(s)]

ygj2n

n-1+nc;r) n .

= (-1) (2i)¹¹{ l:

~(z+S)~r;J

+ cp(z+S)h(s) }dmCS) • j=1 ozJ

lsi

ⁿ

In (6) above \ve no1v get, since

n(n+1)

A ^{= (} -1 )- 2 (n-1)!

(2ni)n

Since the n first integrals all converge as R _, 0 , then so does the last, and \ve get

cp(z) ^{= (-}

~)n(n-1)!{ .~ J

⁰_2J'j(z+S)

e(~~gjdm(s)+ r

cp(z+s)h(s)dm(s)} ⁰

J =1 0n

o

z

I s I

₀

'h

Let h .(z) := (- n1)n(n-1)!

f

^cp(z+s)

p(~)gjdm(s),

and

J -

0

'h lsln

h(S) I= (-

~)

¹¹

(n-1)!h(S)

We then get, (supp p is compact):

n

oh. l

cp(z) ~ E

--'l.

^{(z) + ·} cp(z+s)h(S)dm(s) • j~1

ozJ

C

Here h

1, ••• ,hj E ~(U) , and also g E ~(U) , 1vhere g(z) := j r cp(z+s)h(s)dm(s)

en

This is clear, since supp cp cue and supp p c Usl< e}

(7)

•

If we further assume that supp cp c u2e cu e ' then supp ^g c ue/21$)

In fact, let z E U- Ue _• I f lsl~e, then h(s) ^{~ 0} since p(S) _{~ 0'} and thus cp(z+s)h(s) ~

o •

^{I f} Is

I

< e , then

z + s E U- u2e, and thus cp(z+s)h(S) ⁼ O•h(S) = 0, which proves (8).

Let further 9 E ~(U) be such that 9

=

1 in a nbh. of ~/

2

^•

Then the function

is where

is translation, h(z) v := h(-z) and (9T)z means that

(9)

9T acts w.r.t. z.

That ( 9T, 'f ~;;h)

v

is 1vell defined, follows since supp 9T is compact and the

cF

statement follows as in part I) of the proof of our first lemma. By (8) 1ve have (T,g) = (ST,g),

(T,g) = (eT,g) = (eT,

J

cp(z+s)h(S)&n(s)>

and further en

= (eT,

j

h(s--z)cp(s)&n(s)> =

en

J <eT,'fs~>cp(s)&n(s)

u2e

(10)

We must prove the last equality, and after that we will quickly finish the proof of the theorem. Consider lh(s-z)cp(s)dm(s) as a limit of Riemann sums of the form E h(sa.-z)cp(s

c

0)m(Sa.), where a.

m(Sa.) is the measure of a rectangle Sa. containing sa. • More generally, if D is a differentiation monomial of order p in the

real components of z E C , then we have D ⁿ

J

h(s-z)cp(s)dm(s)

sums converge uniformly

en

z on compact sets, and thus lim(z-> E h(sa.-z)qJ(sa)m(Sa)) a

= (z _,

J

h(s-z)qJ(s)dm(S))

en

in the space ~(On) • By continuity of 9T on this space we get lim(eT, (z _, ~ h(sa·-z)cp(sa)m(sa))>

= (eT,

J

h(s·-z)cp(s)dm(s)) •

en

The left hand side of this equals lim E ((eT)z,h(sa-z))qJ(sa)m(Sa) •

a Since this is a Riemann sum for

J

((eT)z,h(s·-z))qJ(s)dm(S), (by (9) above the integrand is Ceo with

en

support in supp qJ) , 1ve get (eT,

J

h(s-z)qJ(s)dm(s))

= J

\(9T)z,h(s-z))qJ(S)dm(S)

=

en

n

J .

^{c v}

(eT,-rgh)qJ(s)dm(s)

~ J

^(9T,-r

e~)qJ(s)dm(s), en

u2e

for qJ E ~(u

2

^{e) •} Thus (10) above is proved.

By (7) end (10) above we get for qJ E ~(u

2

^e), using the fact that oT/ozj = 0 for j = 1, ••• ,n :

n .

J

(T,qJ) ⁼ (T, E oh ./azJ + qJ(z+S)h(s)dm(s)) j=1 J

0n

=

~

(T,ah./ozj)+ (T,

J

cp(z+s)h(s)dm(s)>

j=1 J

0n

n .

J

=- E (oT/azJ,h.) + (eT, qJ(z+s)h(s)dm(r;)>

j=o J

en

=

J (eT,-rg~)qJ(s)dm(S),

^{and here}

s _, (eT,-rr;~>,

u2e

which is independent of qJ , is a Thus T equals the ~-function

Ceo-function, v

by (9) above.

s _, <

eT, -r gh>

Since this holds for all e > 0 , \ve have that T is a ~-function, ru1d thus holomorphic.

Q,ED.

References:

[ 1] : Dolbeaul t: "Sur la cohomologie des varietes analytiques complexes", C. - R. Acad. Sci. Paris 236 ( 1953) p. 175-177·

[2]: Gunning: "Lectures on Riemann Surfaces", Princeton, 1966.

[3]: Lang: "Differential Manifolds'', Addison-\vesley, 1972.

[4]: Narasimhan: "Analysis on Real and Complex Manifolds", Masson & Cie

I

North-Holland, 1968.

[5]: Serre: "Un theoreme de dualite", Comm. Math. Helvet. 29, 1955, p. 9-26.

[6]: Treves: "Topological Vector Spaces, Distributions and Kernels'', Academio Press. 1967.